Topology of symplectomorphism groups of rational ruled surfaces

Let $M$ be either $S^2\times S^2$ or the one point blow-up ${\mathbb{C}}P^2\char93 \overline{{\mathbb{C}}P}^2$ of ${\mathbb{C}}P^2$. In both cases $M$ carries a family of symplectic forms $\omega_{\lambda}$, where $\lambda > -1$ determines the cohomology class $[\omega_\lambda]$. This paper calculates the rational (co)homology of the group $G_\lambda$ of symplectomorphisms of $(M,\omega_\lambda)$ as well as the rational homotopy type of its classifying space $BG_\lambda$. It turns out that each group $G_\lambda$ contains a finite collection $K_k, k = 0,\dots,\ell = \ell(\lambda)$, of finite dimensional Lie subgroups that generate its homotopy. We show that these subgroups ``asymptotically commute", i.e. all the higher Whitehead products that they generate vanish as $\lambda\to \infty$. However, for each fixed $\lambda$ there is essentially one nonvanishing product that gives rise to a ``jumping generator" $w_\lambda$ in $H^*(G_\lambda)$ and to a single relation in the rational cohomology ring $H^*(BG_\lambda)$. An analog of this generator $w_\lambda$ was also seen by Kronheimer in his study of families of symplectic forms on $4$-manifolds using Seiberg-Witten theory. Our methods involve a close study of the space of $\omega_\lambda$-compatible almost complex structures on $M$.